Multivariable functions are functions like $f(x,y)=xy^2$, functions that have two or more inputs but still only one output.

There exist multivariable limits, where $x$ and $y$ approach a value instead of just $x$.

In multivariable calculus, when writing $\frac{\mathrm{d}}{\mathrm{d}x} f(x,y)$, one does not assume $y$ to be a constant, instead it is assumed that $y$ is a function of $x$.

To indicate that $y$ is a constant, one should use $\frac{\partial}{\partial x} f(x,y)$. These are called partial derivatives.

One also has integrals of multivariable functions. We say that we integrate over a surface in case of a two-variable function. We write this as $$\iint_S f(x,y)$$

If the surface is a rectangle $S: [a,b] \times [c,d]$, and the function is continuous on this rectangle, then Fubini's theorem says that $$\iint_S f(x,y) = \int^b_a\int^d_c f(x,y) \mathrm{d}y \mathrm{d}x$$

This is useful to evaluate these integrals.

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